Yes, every even integer greater than 4 can be expressed as the sum of two prime numbers and three prime numbers. This is a consequence of the Goldbach’s conjecture and the stronger Vinogradov’s theorem.

Goldbach’s conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. Although this conjecture is still unproven, it has been verified computationally for all even integers up to a very large number.

Vinogradov’s theorem, proved in 1937, states that every sufficiently large odd integer can be expressed as the sum of three prime numbers. The bound for “sufficiently large” has been improved over time, and it is now known that every odd integer greater than 7 can be expressed as the sum of three prime numbers.

Combining these two results, we can conclude that every even integer greater than 4 can be expressed both as the sum of two prime numbers (by Goldbach’s conjecture) and as the sum of three prime numbers (by Vinogradov’s theorem).

For example, consider the even integer 10:

– 10 = 3 + 7 (sum of two prime numbers)

– 10 = 3 + 3 + 5 (sum of three prime numbers)

This property holds for all even integers greater than 4, making it possible to represent them as the sum of two prime numbers and as the sum of three prime numbers.